Sample-to-sample fluctuations of electrostatic forces generated by quenched charge disorder

It has been recently shown that randomly charged surfaces can exhibit long range electrostatic interactions even when they are net neutral. These forces depend on the specific realization of charge disorder and thus exhibit sample to sample fluctuations about their mean value. We analyze the fluctuations of these forces in the parallel slab configuration and also in the sphere-plane geometry via the proximity force approximation. The fluctuations of the normal forces, that have a finite mean value, are computed exactly. Surprisingly, we also show that lateral forces are present, despite the fact that they have a zero mean, and that their fluctuations have the same scaling behavior as the normal force fluctuations. The measurement of these lateral force fluctuations could help to characterize the effects of charge disorder in experimental systems, leading to estimates of their magnitudes that are complementary to those given by normal force measurements.